Mass-energy equivalence

Mass–energy equivalence says that rest mass is part of a system’s energy budget. For a system at rest, E₀=mc². In nuclear physics, tiny differences between the total rest masses before and after a process can correspond to energies large enough to measure directly.

Rest energy and total relativistic energy

E=mc² is the rest-energy relation. A moving particle has total energy satisfying E²=(pc)²+(mc²)². Nuclear reactions conserve total energy and momentum together; rest mass alone need not be conserved as a separate quantity.

A lower final rest mass corresponds to energy carried elsewhere. It can appear as kinetic energy, photons or excitation.

Q-value measures the energy balance

Q = (m_initial - m_final)c²

Using atomic mass units: 1 u·c² ≈ 931.5 MeV.

Q>0 means energy is released. Q<0 means energy must be supplied, although momentum conservation can make the required projectile threshold larger than |Q| in some laboratory geometries.

Mass-defect converter

Δm (u)

Binding energy belongs to the whole system. A bound nucleus has lower rest energy than its separated nucleons.

Why c² makes tiny mass differences important

The speed of light squared is an enormous conversion factor in SI units. That is why a mass change far below ordinary weighing precision can correspond to MeV-scale nuclear energies. Everyday chemical binding also changes mass, but the relative changes are much smaller.

Fusion and fission fit the same bookkeeping

Both can release energy when the products are more tightly bound and have lower total rest mass than the initial system. No conservation law is broken: the “missing mass” corresponds to energy carried by the products and radiation.

The mass belongs to the whole system. Heating a sealed object, compressing a spring or exciting an atom increases its total internal energy and therefore its mass by ΔE/c². In ordinary situations the change is far too small to notice, but the same bookkeeping becomes measurable in nuclear reactions. Thinking in terms of the complete system avoids the misleading idea that E = mc² applies only when matter is destroyed.

Worked examples

1. Convert a mass defect

Solution

Δm=0.0020 u gives Q≈0.0020×931.5=1.86 MeV.

2. Energy from SI mass

Solution

For Δm=1.0×10⁻12 kg, ΔE=Δmc²≈9.0×10⁴ J.

3. Sign of Q

Solution

If products have 0.005 u more rest mass than reactants, Q≈-0.005×931.5=-4.66 MeV; energy must be supplied.